Gray images of cyclic codes over F2+uF2 and Z4

doi:10.3778/j.issn.1002-8331.2010.20.023

Computer Engineering and Applications ›› 2010, Vol. 46 ›› Issue (20): 84-85.DOI: 10.3778/j.issn.1002-8331.2010.20.023

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Gray images of cyclic codes over F₂+uF₂and Z₄

LIANG Hua

School of Mathematical Science，Huaiyin Teachers College，Huai’an，Jiangsu 223300，China

Received:2009-12-31 Revised:2010-03-31 Online:2010-07-11 Published:2010-07-11
Contact: LIANG Hua

F₂+uF₂和Z₄上循环码的Gray象

梁华

淮阴师范学院数学科学学院，江苏淮安 223300

通讯作者: 梁华

Abstract

Abstract: Based on the property of Gray map Φ，cyclic codes over F₂+uF₂ and Z₄ are studied.It is proved that a code of arbitrary length over F₂+uF₂ is a cyclic code if and only if its Gray image is a quasi-cyclic code over F₂，and a necessary and sufficient condition for a code of arbitrary length over Z₄ to be a cyclic code is obtained.In particular，the Gray image of a linear cyclic code of length n over F₂+uF₂ is a linear quasi-cyclic code of index 2 and length 2n over F₂，and the Gray image of a linear cyclic code of length n over Z₄ is a quasi-cyclic code of index 2 and length 2n over F₂.

Key words: linear code, cyclic code, quasi-cyclic code, Gray map

摘要： 利用Gray映射Φ的性质，研究了环F₂+uF₂和Z₄上的任意长循环码。证明了环F₂+uF₂上任意长码是循环码当且仅当它的Gray象是域F2上的准循环码，得到了Z₄上任意长码是循环码的一个充分必要条件。特别的，环F₂+uF₂上长为n的线性循环码的Gray象是域F₂上指标为2长为2n的线性准循环码，环Z₄上长为n的线性循环码的Gray象是域F₂上指标为2长为2n的准循环码。

关键词: 线性码, 循环码, 准循环码, Gray映射

CLC Number:

TN911.22

LIANG Hua. Gray images of cyclic codes over F₂+uF₂and Z₄[J]. Computer Engineering and Applications, 2010, 46(20): 84-85.

梁华. F₂+uF₂和Z₄上循环码的Gray象[J]. 计算机工程与应用, 2010, 46(20): 84-85.

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Gray images of cyclic codes over F₂+uF₂and Z₄

F₂+uF₂和Z₄上循环码的Gray象

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